Parallel algorithms for matching and independence problems in graphs and hypergraphs

by 1977- Windsor, Aaron Andrew

Abstract (Summary)

We consider the following problem: Given a greedy graph algorithm
that seems to be inherently sequential, to what extent can
we expect to speed up the computation by making use of additional
processors? We obtain several positive results for particular problems,
showing that it is theoretically possible to parallelize some
greedy graph algorithms to the extent that their parallel running
times are asymptotically much faster than their sequential running
times. Highlights of our results include:
• A simple proof that a simple algorithm of Luby (“the permutation
algorithm”) is an RN C algorithm for finding a maximal
independent set in a graph, and the first known derandomization
of that algorithm.
• The first non-trivial upper bound on the deterministic time
complexity of finding a maximal independent set in a hypergraph
on a PRAM.
• A partial analysis of the permutation algorithm generalized
to hypergraphs, showing that it outperforms the best known
algorithm for finding a maximal independent set in a hypergraph
in the following sense: For hypergraphs of dimension at
least 6, any set of vertices is at least as likely to be added to the
independent set by an iteration of the permutation algorithm
as it is to be added by an iteration of the other algorithm.
• The first N C algorithm for finding a maximal forest in a hypergraph.
• The first N C algorithm for finding a maximal acyclic set in an
undirected graph.